Cross-validation statistics for the mean and coefficient of variation surfaces

Before carrying out the cross-validation statistics, two sets of the interpolated surfaces are illustrated in Figure 3.5Error! Reference source not found. and Figure 3.6, one for the mean and the other for the cov, based on the statistics obtained from the 235 meteorological stations discussed earlier. In the figures, we only included a few typical interpolated surfaces obtained using the IDW, LPI, RBF-CRS, RBF-SWT, KO, KS, Co- KO, and Co-KS, since some of the 14 interpolation methods included in this study provide very similar results or lead to relatively large RMSE as will be discussed in the next section.

Figure 3.5 shows the interpolated surface of the mean of the wind speed. In general the surfaces are relatively smooth. However, there are, and as expected, typical “bulls eye” patterns due to the application of the IDW. Such an effect is not observed for other methods included in the figure. In fact, a smearing effect could be appreciated from the surfaces obtained using the geostatistical methods, resulting in large regions with similar mean of the wind speed. The (over) smearing effect is most significant for the KS and Co-KS. Moreover, the “bulls eye” pattern is much more pronounced for the plot shown in Figure 3.6 for the cov values if the IDW method is used; the use of the RBF-CRS and RBF-SWT retains some of this pattern but to a lesser degree. Again, the use of the geostatistical methods results in much smoother surfaces of the cov of the wind speed. It

40

should be noted that the KO can be an exact interpolator by setting the nugget equal to zero, and a lesser smearing effect is observed in the interpolated surfaces. These surfaces are not presented to reduce the clutter and limit arbitrariness.

To objectively select the interpolation method, the (leave-one-out) cross-validation analysis was carried out for the 14 considered spatial interpolation methods. Statistics from the cross-validation analysis such as the mean of the prediction error (ME) and the RMSE for the deterministic methods are reported in Table 3.1, where the preferred method could be selected based on the lowest RMSE and/or the ME value that is close to zero (i.e., least biased model). The results shown in the table indicate that for the mean of the annual maximum wind speed the preferred method among the considered deterministic methods is the RBF-SWT which is followed by the RBF-CRS. The differences between the estimated ME and RMSE by these two methods are negligible. For interpolating the surface of the cov of the annual maximum wind speed, the preferred method among the considered deterministic methods is the GPI which is followed by the RBF-SWT, RBF-CRS and LPI. The surface of the cov obtained by the GPI is shown in Figure 3.7. A comparison of Figure 3.6 and Figure 3.7 indicates that the application of the GPI results in a very unnatural or artificial looking surface of the cov. This makes the use of the GPI questionable, and it is not considered further below.

The statistics from the cross-validation analysis for the geostatistical methods are shown in Table 3.2. In this case, the reported statistics are the ME, RMSE, the average (kriging) standard error (ASE), the mean of standardized prediction errors (M-SE), and root mean

41

square standardized prediction errors (RMS-SE) (Johnston et al. 2003). A method that results in the lowest RMSE, and leads to RMSE closer to ASE, and/or RMS-SE close to one is the appropiate and preferred method. Based on these considerations and the results presented in Table 3.2, the preferred method to interpolate the surface of the mean of the wind speed is the Co-KO. This preference is closely followed by the KO. For the cov of the extreme wind speed, the preferred method is the KS which is followed by the Co-KS and KO.

By considering the results shown in Table 3.1 and Table 3.2, and based on the RMSE alone, the preferred deterministic methods that are the RBF-SWT and RBF-CRS outperform the preferred geostatistical methods that are the Co-KO and KO if the interpolation of the mean is of interest. For interpolating the surface of the cov of the wind speed, it can be considered that the performances of the RBF-SWT, RBF-CRS, KS, Co-KS and KO are similar.

The preceding analysis indicates that although the preferred interpolation method for different statistics of the annual maximum wind speed differs, in general the RBF-SWT, RBF-CRS, KO and Co-KO can be considered for interpolating the surfaces of both the mean and cov of the wind speed. The use of any of these methods results in relatively consistent RMSE and interpolated surfaces (see Table 3.1 and Table 3.2 and Figure 3.5 and Figure 3.6).

42 3.4.2 Interpolated surfaces for the quantiles of wind speed

Consider that the mean and cov of the wind speed at a meteorological station, denoted by

mvo and vvo, respectively, are given, and the wind speed is a Gumbel variate. The Gumbel distribution is given by (Benjamin and Cornell 1970),







v u a



v

F( )exp exp (  )/ , (3.1)

where F(v) denotes the cumulative distribution function, v is the wind speed, a and u are the scale and location parameters of the distribution. u equals m_vo0.5772a and a equals



6vvomvo



/. The T-year return period value of the wind speed, vTo, is given by,







T



a u

v_To   ln ln(11/ . (3.2)

which can be written as,







T



m v m v m v v vo vo vo vo vo T To 0.5772 ln ln(1 1/ 6 ) , (        . (3.3)

where v_To v_T(m_vo,v_vo) emphasized that vTo is a function of the mean and cov of the extreme wind speed.

Two approaches may be used in developing the surface of the T-year return period value of the wind speed. The first one is to directly interpolate the T-year return period value of the wind speed, vTs, from vTo obtained at the meteorological stations. The second approach is to calculate the T-year return period value using the adopted Gumbel model with the mean and cov interpolated from those at meteorological stations. The former is focused on the spatial variation of the T-year return period value, while the latter concentrates on the spatial variation of the mean and cov of the extreme wind speed that are used to define the probabilistic distribution model of the wind speed at a site of

43

interest. These two approaches may not necessarily lead to the same predicted return periods.

Consider the first approach. The (leave-one-out) cross-validation analysis was carried out for the 14 considered spatial interpolation methods and using the T-year return period value of the wind speed. The obtained statistics from the cross-validation analysis are presented in Table 3.3 for the deterministic methods and Table 3.4 for geostatistical methods considering T= 50, 500 and 1000 years.

Statistics shown in Table 3.3 indicate that the RBF-CRS is the preferred method judged based on the RMSE values. This preference is closely followed by the RBF-SWT. The ME values for these methods are also small as shown in the table. Results presented in Table 3.4 suggest that the preferred method among the considered geostatistical methods is the Co-KO and followed by the KO for all the considered return periods. This indicates that the use of the terrain’s elevation as the covariate improves the spatial interpolation of the surface of the quantile of the wind speed only slightly because the differences between the estimated statistics by considering the Co-KO and KO are relatively small, as can be observed from the table. Also, the comparison of the results shown in Table 3.3 and Table 3.4 indicates that the RBF-CRS and RBF-SWT outperform KO and Co-KO based on the RMSE alone.

A comparison of the interpolated surface of the T-year return period value of the wind speed by using RBF-CRS, RBF-SWT, KO and Co-KO is presented in Figure 3.8 and

44

Figure 3.9. It indicates that these methods lead to similar surfaces, except that more details are provided if the deterministic methods (i.e., the RBF-CRS and RBF-SWT) instead of the geostatistical methods (i.e., the KO and Co-KO) are used.

Now consider the second approach, where the return period values are calculated using the Gumbel distribution with the mean and cov interpolated using the RBF-CRS, RBF- SWT, KO and Co-KO. To assess the adequacy of this approach, we carry out the cross- validation analysis as follows:

1) Withold the mean and cov of the wind speed at one meteorological station, denoted by

mvo and vvo, respectively. Estimate the mean and the cov for the station, denoted by mvs and vvs, using the data from all the remaining stations. If the KO or Co-KO are used the standard errors of the estimates mvs and vvs, denoted respectively by σmvs and σvvs, are also calculated. This step is the same as was done in the section of cross-validation for the mean and coefficient of variation surfaces of the extreme wind speed.

2) Calculate the predicted value (i.e., the mean) and standard error of the T-year return period value of the wind speed, vT, denoted by vTsp and σTsp, respectively. Since these quantities cannot be obtained directly from ArcGIS and the joint probability density function of mvs, vvs, σmvs and σvvs are not available, an approximated method by using the first order second moment approximation based on Taylor series expansion (Benjamin and Cornell 1970) is used. This and Eq. (3.3) leads to,

), , ( vs vs T Tsp v m v v  . (3.4a) since ₂ 2 vo T m v   and ₂ 2 vo T v v  

45 2 2 2 2 2 vvs vo T mvs vo T Tsp v v m v                      . (3.4b)

where the derivatives are evaluated at mvs and vvs.

3) Repeat 2) and 3) for all stations and, calculate the ME and RMSE for each methods, and the ASE, M-SE and RMS-SE for KO and Co-KO based on vTsp and vTo.

The obtained statistics are reported in Table 3.5 and Table 3.6. The results show that based on RMSE alone, the RBF-CRS and RBF-SWT outperform KO and Co-KO, which is consistent with the observation drawn from the first approach.

The comparison of the ME and RMSE values shown in Table 3.3 and Table 3.5 indicates that the second approach outperforms the first approach. However, the differences in the estimated RMSE values are less than about 2%. This implies that the application of one or the other approach for interpolating the return period values is equally adequate and leads to similar statistics.

If the KO and Co-KO are used, the results presented in Table 3.4 and Table 3.6 indicate that based on the RMSE alone, the use of the second approach is preferred. This preference is reversed if the decision is based on the closeness of RMS-SE to unity. Again, in all cases, the numerical differences in the obtained statistics from the cross- validation analysis, including RMSE and RMS-SE, are not large. This implies that the selection of the best approach (approach one or two) and spatial interpolation methods is difficult, and their performances are comparable.

46

3.5

Conclusions

Spatial interpolation of the statistics (i.e., mean and coefficient of variation (cov)) and the

T-year return period value of the annual maximum (hourly-mean) wind speed is needed for site dependent structural reliability analysis and the codification of the wind load. This study is focused on the selection of the preferred method for spatially interpolating the mean, cov, and return period value of the wind speed for Canada. The analysis used both deterministic and geostatistical methods and the wind records at 235 sites with temporal coverage ranging from 17 to 58 years of useable data. The cross-validation analysis results for the surfaces of the mean and cov of the extreme wind speed indicate that:

Overall the RBF-CRS, RBF-SWT, KO and Co-KO are preferred methods among 14 considered deterministic and geostatistical methods. In general and as expected, the deterministic methods result in surfaces with more details or less smoother transition than the geostatistical methods.

Based on RMSE alone, the RBF-CRS and RBF-SWT outperform the KO and Co-KO if the surface of the mean of the wind speed is of concern, and this is reversed if the surface of the cov of the wind speed is of interest.

A cross-validation analysis is also carried out for two possible approaches in estimating the T-year return period value of the wind speed: the first approach spatially interpolates the return period value alone, while the second approach concentrates on the spatial

47

interpolation of the statistical characteristics of the extreme wind speed and its use for estimating the return period values at sites of interest. In general, for both approaches, the RBF-CRS and RBF-SWT outperform the KO and Co-KO. Based on the estimated ME and RMSE values, the second approach is more preferred than the first approach for all the 4 interpolation methods. In all cases, the differences between the estimated ME and RMSE are not very large (see Table 3.3 to Table 3.6).

Based on the above observations, it is recommended that the RBF-CRS or RBF-SWT to be employed for the spatial interpolation of the mean, cov and the T-year return period values if more details on the interpolated surfaces are desired. If an increased smearing effect is preferred, the use of the KO or Co-KO can be adopted.

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Table 3.1 Statistics from the cross-validation analysis for deterministic methods.

Method

Mean of annual maximum wind speed

cov of annual maximum wind speed

ME (km/h) RMSE (km/h) ME (km/h) RMSE (km/h) IDW -2.49E-1 10.19 -2.96E-4 3.27E-2

GPI 3.54E-2 12.11 -1.90E-5 3.07E-2 LPI 5.64E-1 10.22 3.26E-5 3.26E-2 RBF-CRS -2.01E-1 9.961 -2.48E-4 3.20E-2 RBF-SWT -1.99E-1 9.957 -2.36E-4 3.18E-2 RBF-M 7.73E-3 10.10 -4.93E-5 3.51E-2 RBF-IM -2.91E-1 10.06 -2.98E-4 3.18E-2 RBF-TPS 2.58E-2 11.09 1.94E-4 4.15E-2

Table 3.2 Statistics from the cross-validation analysis for geostatistical methods.

Method

Mean of annual maximum wind speed cov of annual maximum wind speed ME (km/h) RMSE (km/h) ASE (km/h) M-SE RMS-SE ME (km/h) RMSE (km/h) ASE (km/h) M-SE RMS-SE KO -0.17 10.32 10.7 -1.3E-2 0.956 -9.86E-5 3.18E-2 3.0E-2 -2.8E-3 1.047 KS -0.42 10.59 12.6 -3.1E-2 0.834 -1.65E-4 3.12E-2 3.1E-2 -5.3E-3 1.002 KU 0.20 12.80 15.7 -2.3E-3 0.874 1.09E-3 3.58E-2 3.6E-2 1.1E-2 1.002 Co-KO -0.15 10.27 10.6 -1.1E-2 0.963 -1.13E-4 3.18E-2 3.0E-2 -3.3E-3 1.046 Co-KS -0.33 10.98 12.3 -2.6E-2 0.887 -1.45E-4 3.12E-2 3.1E-2 -4.7E-3 1.007 Co-KU 0.22 11.65 13.5 -5.9E-4 0.850 1.24E-4 3.67E-2 3.8E-2 -4.9E-3 1.009

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Table 3.3 Statistics from the cross-validation analysis for T-year return period value of wind speed by deterministic methods.

Method

50-year return period value of wind speed

500-year return period value of wind speed

1000-year return period value of

In document Spatial Variation and Interpolation of Wind Speed Statistics and Its Implication in Design Wind Load (Page 66-92)